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1. Report No.
MAC TM-35
3. Recipient's Accession No.
4. Title and Subtitle
An Interactive Implementation of the Todd-Coxeter Algorithm
5. Report Date . Issued
December 1973
7. Author(s)
Richard J, Bonneau
8- Performing Organization Rept.
MAC TM-3.5
9. Performing Organization Name and Address
PROJECT MAC; MASSACHUSETTS INSTITUTE OF TECHNOLOGY :
545 Technology Square, Cambridge, Massachusetts 0213 9
10. Project/Task/Work Unit No.
11. Contract/Grant No.
N00014-70-A-0362-0006
12. Sponsoring Organization Name and Address
Office of Naval Research
Department of the Navy
Information Systems Program
Arlington. Va 22217
13. Type of Report & Period
Covered: Interim
Scientific Report
14.
15. Supplementary Notes
16. Abstracts
The Todd-Coxeter algorithm provides a systematic approach to the
enumeration of cosets of a finitely presented group. This memo describes
an interactive implementation of algorithm, including a manual on its use,
examples, and methods of accessing the program. Applications of this
algorithm are also discussed.
17. Key Words and Document Analysis. 17a. Descriptors
Group Theory
Symbol Manipulation
17b. Identifiers/Open-Ended Terms
17c COSATI Field/Group
18. Availability Statement
Unlimited Distribution
Write Project MAC Publications
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Report)
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UNCLASSIFIED
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26
22. Price
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FORM NTIS-35 (REV 3-72)
USCOMM-DC 14952-P72
MAC TECHNICAL MEMORANDUM 35
AN INTERACTIVE IMPLEMENTATION
OF THE
TODD-COXETER ALGORITHM
Richard J. Bonneau
December 1973
This research was supported in part
by the Raytheon Advanced Degree Program
and by the Advanced Research Projects
Agency of the Department of Defense
under ARPA Order No. 2095 which was
monitored by ONR Contract No. N00014-
70-A-0362-0006.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
PROJECT MAC
CAMBRIDGE MASSACHUSETTS 02139
An Interactive Implementation
of the
Todci-Coxeter Algorithm
Abstract
The Todd-Coxeter algorithm provides a systematic approach
to the enumeration of cosets of a finitely presented group.
This memo describes an interactive implementation of algorithm,
including a manual on its use, examples, and methods of
accessing the program. Applications of this algorithm are
also discussed.
TABLE OF CONTENTS
1 . Introduction 3
2. User's Manual for the Todd-Coxeter Algorithm 4
3. Sample Session Number 1 8
4 . Disk Files for Input and/or Output 10
5 . Sample Session Number 2 11
6. Sample Session Number 3 17
7 . Miscellaneous Information 22
8. Getting onto the MATHLAB Computer 23
Bibliography 24
-3-
1. Introduction
This memo presents a user's manual for the operation of an interactive
version of the Todd-Coxeter algorithm for the computation of the index of
a subgroup with respect, to a given group. The bulk of the FORTRAN program
was derived from the program written by Cannon, et. al [1], but has been
re-organized to provide for interactive use.
The Todd-Coxeter Algorithm, as described in [2], [3] and [4], provides
an efficient method for determining the order of a finite group which is
described in terms of generators and relations. The algorithm also can be
used to compute the index of any subgroup provided the subgroup is presented
by generators which are words in the generators of the group.
This algorithm has been used widely by group theorists and mathematicians
for a number of applications. These applications include construction of a
presentation of a known group by generators and relations [5], computation
of part of the Schur multiplier of a small finite Lie group (PSU(3,4)) [5],
calculating double cosets H x K given the single coset table form of the sub-
groups H and K [6], and determination of a set of coset representatives for
a subgroup [6]. The Todd-Coxeter algorithm has been acknowledged as a power-
ful and necessary tool for any automated group theory project.
The current program is written in the FORTRAN IV programming language
and runs under a simulated PDP-10/50 operating system. The actual configura-
tion is the Project MAC, MATHLAB PDP-10 computer at M.I.T.
This memo presents instructions, and examples, for the use of this pro-
gram, along with an annotated listing of the program. Any comments, criticisms
or suggestions will be graciously accepted and gratefully acknowledged and
should be addressed to the author.
The implementation of the Todd-Coxeter algorithm represents the first
step in an effort to develop a sophisticated collection of programs to aid
in computations with groups. It is our hope to be releasing future memos
detailing improvements and additions to this project.
-4-
2. USER'S MANUAL FOR THE TODD-COXETER ALGORITHM
1. Find a free console on the MATHLAB system.
Type p ! control-Z n 1 (ctrl-Z), ( Non - Printing )
2. The system will type several lines relating to
the number of users, operating system version,
and other information. When it has been printed
out, type
j: LOGIN GROUP ~| followed by a carriage return (CR).
3. TEN50 will be awaiting commands when it types a
new line with only a period on it (.) .
4. To run the Todd-Coxter program type:
| R TC (CR)j
5. At this point, the user has a number of options relating
to the input to the program: input may either
be interactive via the teieypeor automatic via
files on the disk, Similary, the output can be
printed either on the teleype or onto a file on
disk- The remainder of the description will assume
teletype (TTY) for both input and output,
6. The program now prints out the following question:
INPUT ON DSK (20) OR TTY (05) ?
7. Since we desire TTY as input source, we type
[05 rcRjl
8. The question is now printed:
OUTPUT TO DSK (21) OR TTY (05) ?
9. Again, for TTY output we type
J05 (CR]
10. The program now types out the following message:
TODD-COXETER ALGORITHM running on file: TTY. DAT
putting, output on file: TTY. DAT
11. The program now prints out
NEW PROBLEM INPUT
-5-
12. At this point, the program is expecting a control
number to direct its actions. To begin a new
problem (i.e., a group and a subgroup), type
13.
14.
TcrTI
The program will now respond:
PROBLEM n
GROUP RELATORS
where n is the current problem number*
At this point, the group relators are input, using
the following scheme:
For each relator, one line of numbers is input.
Each relator is numerically represented by a
sequence of two digit numbers in the form:
e.x x x (CR)
112 n
15.
where e. is a positive two digit number signifying
the multiplicity of the relator
x. is non-zero integer representing either a
generator of the group (x. positive) or
the inverse of a generator of the group
(x. negative) .
E.g. , if a relator is
«1 -2 3
(aba b ) = 1
then it is entered as the following string:
030102-1-2-2 (CR)
After each line is entered the program will verify the
input by printing (as in the example above) a
EXP 3
1 2 -1 -2 -2
16. Enter as many lines as there are group relators. To
inform the program that all the relators have been
input, enter
r-2 (CR)1
■6-
17, The following line will now be typed:
SUBGROUP GENERATORS
18- You now enter, in the same format as described above,
the generators of the desired subgroup; again,
one line of input per generator,
19. All the generators have been input, enter a
This will inform the program that all the input information
has been given, and also that program execution should
start. (Note: if the identity subgroup is the desired input
you need only type -3 when subgroup generators are asked
for).
20. The Todd-Coxeter algorithm is now being applied to the
input data. The program will now terminate in one
of two ways:
Successful Termination : In this case, the index
has been determined and the following is
printed:
I NDEX= i MAX= m TO TAT> t
Here, i will be the index, m will denote
the maximum number of cosets existing at any
one time during the execution, and t is the
total number of cosets created.
Unsuccessful Termination : In this case, the core
storage was not sufficient to accomodate the
computation necessary to complete the problem.
L00KAHEAD UNABLE TO CONTINUE
21. In both cases, the program again types out
NEW PROBLEM INPUT
22. At this point, the user may wish to proceed in one of
four ways:
1) Input a new problem: enter }j-I (cr)j and proceed
to step 13.
2) Terminate the session: enter 1-9 (CR)J and proceed to
step 2:
3) Maintain the same group as j ust previously input
but input a new subgroup :^' enter -4 (CRn
and proceed to step 17,
4) Maintain the session, W_ ™ th new I/O configuration:
enter _ [~-8 (CR)|
and proceed to step §»
23. At the end of the session, the system types out certain
run- time statistics, then types
24.
EXIT
tc
To log off t he system, type j|7cntrl-Z)l and then
r?T 5gou"t I (CR)
3 . Sample Session Number 1
U
10556) .SLEEP 17, SOu
ET.'T'rS 811 CONSOLE 7 FREE. 16:46:22
EL ITS. 811. DDT. 51 6-
11. USEES
■ETERIEEDTAL SYSTEE WITH DYRAEOD MAGTAPE RCUTIGRS,ETC
LED CAUTION, ANP DOET RELOAD OLD SYSTEM V.TTROUT E^EIi^G k,:,. —rr
Po-Pn rrouv,
j. i> i. j.
*:TEE50
E.PUT Or. DSK (20) 01: TTY (05)
OUTIUT TO DSK (21) OR TTY (05) ?
TODD-COXETER ALGORITHM EUEKIEG OR RILE: TTY .DAT
PUTTING OUTPUT OR PILE: TTY .DAT
IRE-; PEOPLED IRPIT'
-1
-'■•'-n-T T v ' i
GROUP RELATORS a
9 1 OL =. H
<XP 9
*»
8 2
1 2-1-2 A.bflL J b"'c I
EXP 1
-2
-1 -2
SUBGROUP GEXEEATOR
JELEX=
'('/
h;-,X=
\2:
T01AL= AW
NEE PEOELEX II3PUT
_c
EXECUTION TIME: 4.06 SEC.
TOTAL ELAPSED TIRE: ., #,_-,^ ^-' C '
TO EXECUIICE EXPOSE ii^:i'A,iii.
EXIT
-10-
4. Disk Files for Input and /or Output
The current implementation of the Todd-Coxeter algorithm
provides a number of input-output media. In particular, the
teletype (TTY) has been used for both input and output in the
sample session. This section will be concerned with the use
of disk files for both input and output.
Naming Conventions
Disk files used for the FORTRAN I/O must agree with the
following rules:
1) Each file name is of the form
Name- ex t
2) Name must be a 5 character word
3) Ext must be the 3 characters DAT
4) For standard (i.e. default) names are:
INPUT: F0R2O.DAT
OUTPUT: F0R21.DAT
The user is permitted to use standard file names or provide
specific I/O file names.
Disk Files for Input
The structure of these files mirror exactly the inputs
as given in the algorithm.
A sample session with disk input from file NEWCN. DAT and
output to TTY is given below, along with listing of
NEWCN. DAT.
-11-
5 . Sample Session Number 2
EMUI 01; DSK (20) OR TTY (05) ?
20
GTAKDAED FILE BAKE (20) OR LOT (22) ?
22
ALTER 5 LETTER JTLE NAME
BE WCA\ AXE
OUII-UT 10 DSK (21) OR TTY (05) ?
05
T0BB-C0XETER ALGORITHM KUNNIIJG OR TILE: KERUERDA:
TUTTIRG OUTPUT OR PILE: TTY .DAT
**-x-* **************************
PROBLEM 2
GROUP RELATORS
JL.Ai."'
8
1
iE-Ax'
7
2
F;XP
2
i
2
E,aE
./
_-i
O
ou I'O itOUiv Or.:i:r.iLHj. Oiuj
EXP 2
EXT 1 -12
INDEX= 448 MAX= 2176 T0TAL=
******************************
■12-
PROBLEM 3
GROUP RELATORS
EXP
4
3
EXP
5
2
EXP
1
2
2 -
EXP
2
2
3
EXP
1
1
1
EXP
3
1
2
EXP
11
1
EXP
SUBC
ROUP
1
OEKEE/
1
\TGRS
1 1-2-1
I 1 j> 3
EXP 1 1
OVERFLOW MAX= 2740 TOTAL= 3U95
720 COSETS REMAIN AFTER LOOKAKEAD
INDF:X= 720 MAX= 2740 TOTAL= 35S5
PROPLEH 4
GROUP RELATORS
EXP
3
1
EXP
2
1
2
EXP
2
2
■z
->
EXP
2
3
1
EXP
2
1
2
EX?
7
2
EXP 13
SUBGROUP
3
GENERA
TO
?s
•13-
oveeflow max= 2744 tctat- 3710
20c5 cosets remain after lookahead
ovfmflow max= 2744 total- 4477
25^7 cosets remain after lookahead
overflow kax= 2744 iotal=- 4704
2675 cosets remah: aftfe lookahead
overflow max= 2744 total- 4776
10^2 cosets rei-ain after lookahead
irdrx= 1092 max= 2744 tctat- 4776
PEOPLES 5
GROUP REJATORS
EXP
3
EXP
•^
>
EXP
3
EXP
3
O v,
EXP 4 1 3
1 -3
-;-■■■ vn
4
EXP
4
f.;xf
1
EXP
5
-2 3
1-2 1 2-3 1 3 1 -3
1 2
EXP 5 -1 2
SUBGROUP GENERATORS
EXP 1
EXP 1
OVERFLOW HAX= 2739 TOTAL- 3394
2287 COSETS REMAIN AFTER LOOKAHEAD
OVERFLOW MAX= 2739 TOTAL= 3S : 48
2539 COSETS REMAIN AFTER LOOKAHEAD
■14-
OVERFLOW MAX= 2739 TOTAIj= 4052
2702 COSETS REMAIN AFTER LOOK AHEAD
OVERFLOW MAX= 2739 TOTAL= 4089
2726 COSETS REKAIN AFTER LCOKAHEAD
OVFRFIOU MAX= 2739 TOTAL= 4102
2722 COSETS REKAITi AFTER LCOKAHEAD
OVEEFIOlv HAX= 2739 TOTAL= 41 1S
2738 COSETS REIAIK AFTER LCOKAHEAD
OVERFLOW MAX= 2739 TOTAL= 4120
2739 COSETS EEEAIN AFTER LGOKAIFAD
LCOKAHEAD UNABLE TO CONTINUE
■15-
■ Tnj: 2E7C2.LA2
-1
r ■
O I
7 2
2 1 2
7-1 2
2 1
1-1 2
-1
3 -
■A -l -^ -' "1 -A
11
1 1
rf-:: t t_
7 2
y <-
c _j
4 i-o
7-1 7
1 7 1-
1
1
: 8
■16-
Disk Files for Output
The structure of these files mirrors exactly the output
as would be obtained using the TTY for output. In this made,
however, summary information is output to the TTY. A sample
session with input from file NEW CN.DAT and output to file
OUTCN.DAT is given below, as well as the type - out of
OUT CN.DAT.
•17-
6. Sample Session Number 3
4237) .10! 1,16 $$u
ML ITS 811 CONSOLE 7 FREE. 17:01:31
ML ITS. 811. DDT. 51 6.
12. USERS
THE 110 BAUD NUMBER IS 258-7894. SORRY ABOUT THE MIX UP
— PJ
tlo/rin group
INIT
*:TEN
.R TC
INPUT Oli DSK (20) OR TTY (05) ?
2CJ,
STANDARD FILE NAME (20) OR NOT (22) ?
22
ENTER 5 LETTER 1ILE NAME
liEWCN
OUTPUT TO DSK (21) OR TTY (05) ?
STANDARD PILE NAME (21) OR NOT (23) ?
23
ENTER 5 LETTER FILE NAME
OUTCN
TODD-COXETER ALGORITHM RUNNING ON FILE: NEWCN.DAT
PUITING OUTPUT 1 ON FILE: OUTCN.DAT
****■*-*************************
PROBLEM 1
INDEX= 448 MAX= 2176
******************************
PROBLEM 2
TOTAI- 2626
INDEX= 720 MAX= 2740
******************************
PROBLEM 3
TOTAL= 3995
INDEX= 1092 MAX= 2744
**************** **************
TOTAL= 4776
■18-
PROBLEM 4
LOOKAHEAD UNABLE TO CONTINUE
IliPUT OK DSK (20) OR TTY (05) ?
TY OUTCN.DAT
***.************ ***************
pROBBEi; 1
GROUP RELATORS
EXP
C
1
EXP
7
2
EXP
2
1
2
EXP 3 -1 2
SUBGROUP GENERATORS
EXP 2
1
-1 2
INDEX= 448 KAX= 2176
*^***************************
PROBLEM 2
TOTAL= 2626
GROUP RELATORS
EXP
4
EXP
5
2
EXP
1
2
2
-3
-2
3
EXP
2
2
3
3
EXP
1
1
1
1
1
-2 -1
2
EXP
3
1
2
3
EXP
11
1
■19-
SEHPR05P GENERATORS 1 1 3
EXP 1 1
OVERFLOW MAX= 2740 TOTAL= 3995
720 COSETS REMAIN AFTER LOOK AHEAD
INDEX= 720 MAX= 2740 TOTAI- 3995
PROBLEM 3
GROUP RELATORS
VYT
3
1
LVP
2
1
2
EXP
2
2
"a
>
EXF
2
3
1
EXP
2
1
2 3
EXP
7
2
EXF 13
SUBGROUP
3
GENERATORS
OVERFLOW MAX= 2744 TOTAL= 3710
2065 COSETS REMAIN AFTER LOOKAHEAD
OVERFLOW MAX= 2744 TOTAL= 4477
2527 COSETS REMAIN AFTER LOQKAHEAD
OVERFLOW RAX= 2744 TOTAL= 4704
2675 COSETS REMAIN AFTER LOOKAHEAD
OVERFLOW MAX= 2744 TOTAL= 4776
1092 COSETS REMAIN AFTER LOOKAHEAD
INBEX= 1092 MAX= 2744 TOTAL= 4776
it***-*********-*****************
-20-
PRGBLEK 4
GROUP IJiLATORS
AjI-Aj i i"w' : - i' W-* -'< i';l'-i !: ^
w V i J -X- .: a.,'W
-2L
>2 CCi.ll
;.'T r <
ALCA'
a?-;aa; afte;:
.A.
\V_ AT
20
T.U.r.Aii-^AD Li-jAi-.J..
1 .!. J 'i o.l:
-22-
7. MISCELLANEOUS INFORMATION
Typing Errors
1. Before carriage return
2. Use "RUBOUT" or "DELETE" Keys
3. Will echo the deleted character
4. May be used in succession to delete several char.
Program Interrupt :
1. Type control -C (f c) once
2. Until EXIT
ft
appears
To continue after t C interrupt :
1. Type CONT
2. Back to same spot!
To Restart
1. Type START at monitor level
2. Continue as normal
Current Limitations
1. Number of generators ^ 9.
2. Number of terms in generators or relation words ^ 40,
-23-
8. Getting onto the MATHLAB Computer
In order to run the Todd-Coxeter Algorithm on a given problem, one must
first gain access to the MATHLAB computer located at M.I.T. There are three
ways of doing so:
1) If you are at M.I.T. and have access to a free ML console,
simply follow the instructions in Section 2 of this memo.
2) If you are not at M.I.T., but have access to a computer on
the ARPA network of computers, then you can ur 2 the MATHLAB
computer by accessing it over the network. Consult your
local expert on the ARPA network to find out how to do this.
3) You can access the MATHLAB computer using a terminal (teletype
or Memorex) by telephone, If you have an acoustic coupler or
a Dataphone. The telephone numbers are: available from Dr.
Vera Pless or Richard Bonnea^, telephone number (617) 253-6026.
-24-
BIBLIOGRAPHY
[1] Gannon, et al., "Implementation and Analysis of the Todd-Coxeter Al-
gorithm", to appear in Math. Comp .
[2] Coxeter & Todd, "A Practical Method for Enumerating Gosets of a Finite
Abstract Group", Proc. Edinburgh Math. Soc , (2), 5(1936), 26-74.
[3] J. Leech, "Coset Enumeration on Digital Computations", Proc Comb. Phil.
Soc , 59(1963), 257-267.
[4] H. Trotter, "A Machine Program for Coset Enumeration", Canadian Math. Bull.
Vol. 7, No. 3, July 1964, pp. 357-368.
[5] Grover, et al . , "Applications of Coset Enumeration", 2nd Symposium on
Symbolic and Algebraic Manipulation , ACM, March 1971, pp. 183-187.
[6] J. Cannon and G. Hovas , "Applications of the Todd-Coxeter Algorithm",
Technical Report No. 9, Computer Aided Mathematics Project, Department
of Pure Mathematics, University of Sydney, Sydney, Australia, June 1973.
MIT/LCS/TM-35
AN INTERACTIVE IMPLEMENTATION
OFTHE
TODD-COXETER ALGORITHM
Richard J. Bonneau
December 1973
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
PROJECT MAC
Reply to: Projecf MAC
545 Technology Square
Cambridge, Mass. 02139
Telephone: 3 ~5 856
PUBLICATIONS TR/TM FORM
TITLE OF THESIS OR REPORT: An Interactive Implementation of
the Todd-Coxeter Algorithm
AUTHOR(S) Richard J. Bonneau
NO. ASSIGNED: MAC TM-35
Do you recommend it be made into a Project MAC technical
report? If so, what type?
Technical Report (formal, widely distributed)
_x Technical Memo (informal, limited distribution)
Do not print as a MAC publication
i i
i ■ f
? /// \ — . ? ?
(/ .* A: r j
Signature of M^C Group Leader
^ *^3r 5 , 1 °t 7 X
Date
PUBLICATIONS DISTRIBUTION
PROJECT MAC, ROOM 417-A
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
545 TECHNOLOGY SQUARE
CAMBRIDGE, MASS. 0213 9
December 1973
We have recently issued Project MAC Technical Memorandum TM-35:
An Interactive Implementation of the
Todd-Coxeter Algorithm
Richard J. Bonneau
AD 770-565
ABSTRACT
The Todd-Coxeter Algorithm provides a systematic approach
to the enumeration of cosets of a finitely presented group.
This memo describes an interactive implementation of algorithm,
including a manual on its use, examples, and methods of
accessing the program. Applications of this algorithm are also
discussed.